A homogeneous Poisson process, starting at time t = 0 in which events occur at random times with rate> 0. Let X(t) denote the number of events occurring in the interval (0, t] and suppose 0 < t₁< t₂.

1. Are the random variables X(t₁) and X(t₂) independent? Justify your answer.

2. Suppose that exactly one event has occurred in the interval (0, t₂] . Let T₁denote the time that this event occurred. Show that T₁~ U(0, t₂] i.e show T₁has a uniform distribution on the interval (0, t₂) .

hint: if T~U(a,b) then P(Tt) = (t - a) / (b - a) for t(a, b] , 0 < a < b